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Modelling Simul. Mater. Sci. Eng. 7 (1999) 909–928. Printed in the UK PII: S0965-0393(99)07765-7
The relation between single crystal elasticity and the effective
elastic behaviour of polycrystalline materials: theory,
measurement and computation
J M J den Toonder†, J A W van Dommelen‡ and F P T Baaijens‡
† Philips Research Laboratories, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands‡ Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands
Received 8 April 1999, accepted for publication 1 September 1999
Abstract. Due to continuing miniaturization, characteristic dimensions of electroniccomponentsare now becoming of the same order of magnitude as the characteristic microstructural scales of the constituent materials, such as grain sizes. In this situation, it is necessary to take into accountthe influence of microstructure when studying the mechanical behaviour. In this paper, we focuson the relation between the (anisotropic) properties of individual grains and the effective elasticbehaviour of polycrystalline materials. For large volumes of materials, the conventional averagingtheory may be applied. This is illustrated with experiments on various barium titanates. For smallvolumesof material, we examine therelationshipby means of micromechanical computationsusinga finite-element model, allowing the simulation of a real microstructure, based on a microscopicimage of the grainstructure. Various cubic and tetragonalmaterialsare studied. The computationalresults clearly show the influence of the specific microstructural properties on the effective elasticbehaviour.
1. Introduction
The classical way to model the mechanical response of a structure to a certain load is to
use continuum mechanics theory. This theory assumes that the constituent materials are
homogeneous, and in most cases isotropic, without explicitly considering the microstructure of
the materials. This approach is justified as long as the structure has characteristic dimensions
much larger than the microstructural features of the materials such as grain sizes. This will
indeed be the case for most engineering structures, such as steel bridges or concrete buildings.
However, the continuum approach breaks down if the characteristic dimensions of the structure
are of the same order of magnitude as the characteristic microstructural scales. Due to the
distinct trend of miniaturization in electronic component technology, we have now indeedapproached the point where this is the case. An example is the trend in development of the
multilayer ceramic capacitor (MLCC).
The MLCC, of which a schematic illustration is represented in figure 1, consists of stacks
of ceramic dielectric layers interspersed with thin metal electrodes. The ceramic materials are
commonly based on polycrystalline barium titanate (Hennings [1]). Typical thicknesses of
ceramic and metal layers in present-day MLCCs are 10 and 2 µm, respectively. The electric
capacity per volume of an MLCC is enhanced by reduction of dielectric layer thickness.
Figure 1. Left: schematicrepresentation of a multilayer ceramic capacitor(MLCC).Right: ceramiclayers in an MLCC. The metal electrodes are plates containing holes, so in an intersection theyappear as interrupted lines.
Currently, there is a development to reduce the ceramic layer thicknesses to less than 3 µm
and internal electrodethicknessesto less than 1 µm, whereas thenumber of stacksmay increaseto up to 500 layers.
The consequence of this tendency of miniaturization is that the individual ceramic layers
can no longer be considered as continuum materials. Since thelayer thickness is reduced nearly
to grain sizes, the layers will consist of only a few grains in the out-of-plane direction. Figure 1
shows a practical example of this; in this case, the ceramic layer thickness even virtually
equals the average grain size. The structural integrity of these layers will be dominated by the
properties of the individual grains.
It is clear that the continuum approach will fail to provide an accurate description of
the mechanical behaviour of the MLCC and also of other micro-electronic components, and
the microstructure of the material must be accounted for explicitly. The relatively new field
of micromechanical modelling of materials, or, in short, micromechanics, does exactly this.
In micromechanics, materials are modelled on a microstructural scale that is larger than the
atomic scale, but smaller than the continuum level. In polycrystals, the microstructural features
considered act typically on the scale of the grains. The ultimate goal of micromechanics is
to find the relation between the microstructural properties of a material and its macroscopic
behaviour.
In the past few years there has been an increased interest in micromechanics. This has
originated from the demand to be able to design a material with certain favourable properties
by manipulating its microstructure, and it has been stimulated by the increased miniaturization
of components, as discussed above. Examples of recent micromechanical studies are the
modelling of brittle intergranular failure in microstructures (Grah et al [2]) and computation
of effective elastic constants of polycrystalline thin films (Mullen et al [3]).
The aim of the present paper is to make a contribution to the field of micromechanics by
studying the influence of the elastic properties of the separate grains on the effective elastic
behaviour of polycrystalline materials. This was also the focus of Mullen et al [3], but ourapproach has several new aspects. First, we couple analytical, experimental and computational
results. Second, our calculations are based on a real microstructure instead of an artificial
one. Finally, we also consider the tetragonal crystal structure in addition to the cubic crystal
structure. In particular, we consider barium titanate. This material has a cubic crystal structure
above its Curie temperature of about T c = 130 ◦C and a tetragonal structure below T c.
The plan of the paper is as follows. We begin by reviewing the existing theory that
links the microstructural properties to the macroscopic behaviour, which is valid only for
Single crystal elasticity and effective elastic behaviour 911
large volumes of material. This theory is subsequently illustrated with measurements on
various barium titanates over a range of temperatures. The last part of the paper is devoted to
micromechanical computational modelling of thin sheets of polycrystalline materials, as are
present in the MLCCs. The computational results clearly show the influence of the specific
microstructural properties on the effective elastic behaviour.
2. Macroscopic effective constants of large volumes of polycrystalline materials: theory
A sufficiently large volume of untextured polycrystalline ceramic material can be
macroscopically described as an isotropic elastic material that is characterized by a set of two
independent material parameters. However, microscopically, the behaviour of the individual
grains will in general be anisotropic. In this section, we will discuss the elastic behaviour
of individual cubic and tetragonal crystal structures. Both structures exhibit orthotropic
behaviour. Furthermore, we will give a short review of the theory that links the microscopic,
anisotropic properties to the macroscopic, isotropic elastic behaviour.
2.1. Linear isotropic elasticity
Elastic materialsare characterized by a directrelationbetween thelocal stressstateand thelocal
strain state. In the case of linear elasticity this relation can be represented by the generalized
formulation of Hooke’s law, which states that the stress is proportional to the strain
σ = 4C : E (1)
with σ the Cauchy stress tensor and
E = 1
2( ∇ u + ( ∇ u)T) (2)
the infinitesimal strain tensor, where u is the displacement in the material. Equation (1) is
commonly written in a matrix notation. Due to symmetry of the stress tensor and the strain
tensor only six state variables need to be evaluated. Thus, the stress–strain relation is written
as (Nye [4], Ting [5])
σ ∼
= Cε∼
(3)
with
σ ∼
T= [ σ 11 σ 22 σ 33 σ 23 σ 31 σ 12 ] (4)
and
ε∼
T= [ ε11 ε22 ε33 2ε23 2ε31 2ε12 ]. (5)
For general anisotropic elasticitythe elasticity matrixC willbe symmetricand therefore contain
21 independent components.
It is well known that the behaviour of isotropic linear elastic materials can be described
with a set of two independent parameters, for example Young’s modulus E and Poisson’s ratioν. In terms of these parameters, the elasticity matrix C for isotropic materials reads
Alternatively, isotropic linear elastic behaviour can also be characterized with the shear
modulus G in combination with the bulk modulus κ . The sets of material parameters (E, ν)
and (G, κ) are related as
G =
E
2(1 + ν) κ =
E
3(1 − 2ν) . (7)
2.2. Crystal elasticity
Single-crystal elasticity will in general not be isotropic. The number of independent material
parameters depends on the level of symmetry of the crystal structure. For the cubic structure,
the elasticity matrix can be written in terms of three independent material parameters (Nye [4],
Ting [5]):
C =
c11 c12 c12
c12 c11 c12
c12 c12 c11
c44
c44c44
. (8)
If the relation 2c44 = c11 − c12 is satisfied, the material will be isotropic and the elasticity
matrix can be written in the form of equation (6).
Tetragonal crystal structures have lower symmetry properties. As a result, for a complete
description of the constitutive behaviour six independent parameters are needed:
C =
c11 c12 c13
c12 c11 c13
c13 c13 c33
c44
c44
c66
. (9)
In this study, we are particularly interested in barium titanate (BaTiO3
) at various
temperatures. At temperatures above approximately 130 ◦C, which is the Curie temperature
for this material, the barium titanate crystals have a cubic structure. At lower temperatures
the crystal structure is tetragonal. In addition to barium titanate, we consider two more
ceramic materials, namely indium (In) and zircon (ZrSiO4). In table 1 the various independent
components of the elasticity matrix are given for the materials investigated (Hellwege [6],
Berlincourt and Jaffe [7]).
2.3. Effective elastic constants
Generally a ceramic material will consist of a number of grains, each with a different and
unique orientation with respect to a reference frame. All individual grains will microscopically
show anisotropic material behaviour that is dependent on the crystal structure and orientation.
However, when the number of grains is sufficiently large and the orientations are randomlydistributed, the effective macroscopic behaviourwill be isotropic and the constitutive properties
can be characterized by an effective Young’s modulus and an effective Poisson’s ratio.
The macroscopic effective elastic constants are found by averaging the anisotropic elastic
properties of the individual crystals over all possible crystal orientations. To this end a rotation
tensor between a reference frame and a rotated vector basis attached to the crystal structure is
Single crystal elasticity and effective elastic behaviour 913
Table 1. Elastic constants (GPa).
BaTiO3 BaTiO3 In ZrSiO4
cubic tetragonal tetragonal tetragonal
c11 173 275.1 45.2 424
c12 82 179.0 40.0 70
c44 108 54.3 6.52 113
c33 — 164.9 44.9 490
c13 — 151.6 41.2 150
c66 — 113.1 12.0 48.5
The components of the fourth-order elasticity tensor with respect to the rotated frame are
obtained by
4Cijkl =
4Cmnop RlpRkoRj nRim . (11)
The macroscopic effective elastic constants are obtained by averaging this tensor over all
possible rotation tensors (Hearmon [8]):
4Cijkl =
all possible R
4CmnopRlpRko Rjn Rim. (12)
This average is called the Voigt average and for orthotropic materials (see Cook and Young
[16]) it results in the following macroscopic effective elastic constants:
EV = (A − B + 3C)(A + 2B)
2A + 3B + CGV =
A − B + 3C
5 νV =
A + 4B − 2C
4A + 6B + 2C(13)
with
A = c11 + c22 + c33
3 B =
c23 + c13 + c12
3 C =
c44 + c55 + c66
3 . (14)
Another approach is to average the inverse of the elasticity tensor, i.e. the fourth-order
compliance tensor 4S, over all possible orientations
4S ijkl =
all possible R
4S mnopRlpRko Rj nRim (15)
which results in the so-called Reuss effective constants:
ER = 5
3A + 2B + C GR =
5
4A − 4B + 3C νR = −
2A + 8B − C
6A + 4B + 2C (16)
with
A=
s11 + s22 + s33
3 B
= s23 + s13 + s12
3 C
= s44 + s55 + s66
3 . (17)
The Voigt averaging method assumes the strains to be continuous, whereas the stresses are
allowed to be discontinuous. As a resultthe forcesbetween thegrains will notbe in equilibrium.
This method will give upper bounds for the actual effective elastic constants (E,G). When the
Reuss averaging method is applied, the stresses are assumed to be continuous and the strainscan be discontinuous. Consequently, the deformed grains do not fit together and lower bounds
for the effective elastic constants are found.
The Voigt–Reuss–Hill approach combines the upper and lower bounds by assuming the
average of the Voigt and the Reuss elastic constants to be a good approximation for the actual
Table 2. Macroscopic effective elastic constants (GPa), (—), according to the conventionalaveraging procedures.
BaTiO3 BaTiO3 In ZrSiO4
cubic tetragonal tetragonal tetragonal
EV 199.8 162.2 16.8 305.6
GV 83.0 59.8 5.87 119.4
νV 0.204 0.355 0.434 0.279
ER 173.2 130.0 10.6 256.9
GR 69.7 47.6 3.65 98.1
νR 0.243 0.367 0.456 0.310
EVRH 186.5 146.1 13.7 281.3
GVRH 76.3 53.7 4.76 108.8
νVRH 0.221 0.360 0.445 0.293
The macroscopic effective elastic constants of the materials that will be the subject of
investigation in section 4 are represented in table 2. The values are obtained by substitution of
the constants in table 1 into the foregoing expressions.
3. Macroscopic effective elastic constants of barium titanate: measurements
As an illustration of the theory summarized in the previous section, in this section we present
measurements of the macroscopic effective elastic constants of a number of barium titanates
as a function of temperature. The barium titanates are being used, or are considered for use, as
dielectrics in multilayer ceramic capacitors (MLCCs). Owing to the phase transformation
that barium titanate undergoes at the Curie temperature (T c = 130 ◦C), the relationship
between macroscopic effective behaviour and elastic properties of the individual grains is
clearly demonstrated within the temperature range of our measurements, which is 20–400 ◦C.
3.1. The experimental technique and set-up
We determined the elastic constants with the pulse excitation method. The experimental set-up
is sketched in figure 2. A disc-shaped barium titanate specimen is placed on four ceramic balls
on a sample holder. During a measurement, a graphite projectile hits the specimen, causing
it to vibrate with a certain natural frequency. As is well known, the value of the frequency
depends on the specimen geometry, the mass of the specimen, the elastic constants of the
material and the mode of vibration. The latter can be influenced by the way in which the
specimen is supported and by the exact position at which the projectile hits the specimen. In
our experiment, the frequency is captured via a waveguide by a microphone and the signal
is subsequently analysed by a commercially available apparatus (Grindosonic). The whole
set-up is placed in an oven, so that measurements can be made up to 400 ◦C.
We measured two natural frequencies of the specimens, namely that of the fundamental
flexural mode f f (one nodal circle) and that of the fundamental torsional mode f t (two nodaldiameters). Since we know the geometry, the mass and the vibrational modes, we can compute
the Young’s modulus and the Poisson’s ratio of the materials from the two resonant frequencies
measured. For the details, we refer to Glandus [9]. The measurements were conducted over
the temperature range 20–400 ◦C. The test specimens used were circular discs with diameters
ranging from 44.80 to 45.60 mm, and thicknesses between 2.04 and 2.14 mm. The materials
used were five barium titanates; all these materials are BaTiO3 based, each having specific
chemical substitutions (or ‘dopants’), the function of which is to tune the dielectric properties
Figure 4. The temperature dependence of the macroscopic effective Young’s modulus E (left) andPoisson’s ratio ν (right) measured for various barium titanates (curves). The markers indicate thepresent theoretical estimates for pure polycrystalline barium titanate (table 2, VRH averages).
both quantities around the Curie temperature T c = 130 ◦C of barium titanate. It is obvious
from figure 4 that the transition in crystal structure which occurs at T c has a direct influence
on the elastic properties.
Figure 4 also shows that the elastic properties are virtually constant above the transition
temperature. Below the sharp transition, however, the properties still change with temperature.
The Young’s modulus, for example, increases with increasing temperature below 120 ◦C.
This is probably caused by the fact that the dopants present in the materials result in regions
with a cubic crystal structure even below 120 ◦C (i.e. the dopants smear out the global Curie
temperature towards lower temperatures). As a final observation from figure 4, we note that
the Poisson’s ratio shows some fluctuations; it appears that the value ν is sensitive to small
fluctuations in the resonant frequencies measured, which are caused by measurement noise.
The qualitative tendency we found for E is consistent with results reported by Duffy et al
[10] for another MLCC barium titanate.
3.3. Comparison with theory
The theoretical Voigt–Reuss–Hill estimates of table 2 for cubic and tetragonal barium titanate
are plotted in figure 4 along with the experimental results. The theoretical values are
qualitatively consistent with the measurements, i.e. they also exhibit a lower Young’s modulus
at temperatures below T c (tetragonal structure) than above T c (cubic structure). The theoretical
values for Poisson’s ratio match the measurements well. Hence, we may conclude that the
general behaviour of the measurements can be explained primarily by the change in the crystal
structure of individual grains as a function of temperature.
However, a quantitative comparison of the theoretical and experimental results showsthat there is some discrepancy, especially for the Young’s modulus at room temperature. The
explanation for this may be that the theoretical values concern pure barium titanate, whereas
the measurements were conducted with doped BaTiO3. This would also explain why in the
experiments the Young’s modulus increases even at low temperatures, i.e. below T c: the doped
polycrystalline barium titanates at these lower temperatures actually consist of a mixture of
tetragonal and cubic crystals, with the contribution of the cubic structures increasing as the
Single crystal elasticity and effective elastic behaviour 917
4. Effective elastic behaviour of small volumes of various polycrystals: simulations
In the previous sections we saw how the macroscopic effective elastic behaviour of large
volumes of polycrystalline materials can be theoretically related to the anisotropic elastic
properties of the individual grains. This relationship was also illustrated with measurementsof cubic and tetragonal barium titanate. However, when the volume of material is small, so that
it consists of a limited number of grains, the overall elastic behaviour may be highly dependent
on the specific elastic properties of the individual grains. This means that their influence on the
effective elastic properties of the material may not be averaged out, so that the theory discussed
in section 2 is not applicable.
Considering the trend towards miniaturization in many applications (e.g. microelectronic
devices), this situation already occurs in practice. For that reason, we present in this section
numerical simulations of the effective elastic behaviour of a limited aggregate of grains with
various (anisotropic) elastic properties. The results are related to the theory we discussed
above. The finite-element simulations we present here may be considered a first step towards a
more comprehensive modelling of the general relation between the microstructure of a material
and its overall effective mechanical properties.
4.1. Mesh generation
We used a finite-element model (FEM) to compute the elastic behaviour of a microstructure.
The first step in such modelling is the generation of a computational mesh. As the starting
point for the generation of the mesh we took a digitized image of the actual microstructure of
a polycrystalline material. This image is depicted in figure 5, and it represents a microscopic
enlargement of the grain structure of aluminium oxide (Al2O3). The image contains 56 grains
with a typical grain size of 60 µm and consists of 649 × 489 pixels with a grey-value in the
range 0–255.
The LEICA QWin image processing package [11] was used to process the image. Grain
boundaries were detected, and the image was converted to a binary image in which grain
boundaries were represented by zero-valued pixels.
In order to extract from this image a limited set of globally numbered points and curves
which can serve as an input for a mesh generator, a series of operations was performed in
MATLAB [12]. Figure 6 shows the result, which consists of a set of globally numbered points
representing the corners of the grains; a set of globally numbered curves representing parts
of grain boundaries, each of which is described by the numbers of the sequential points it
connects; and, finally, a set of surfaces, each of which represents a grain. The numbers and
directions of the curves describing the grain boundary are known for all surfaces.
Finally, a finite-element mesh was generated with use of the SEPRAN mesh generator
[13]. Because of the irregular shapes of most grains, the surfaces were meshed with triangular
elements. All points defined in the input for the mesh generator appeared as nodal points of
the finite-element mesh. The result was a complete mesh topology description of the total
grain structure. For each individual grain the set of elements was available and could be givenspecific anisotropic material properties. In figure 7 the finite-element mesh for the aluminium
oxide grain structure, consisting of 3996 elements, is represented.
4.2. Numerical procedure
We carried out various FEM calculations on the mesh in figure 7, using the FEM package
MARC [14]. Details of the element formulation are given in appendix B. We considered
Single crystal elasticity and effective elastic behaviour 919
Figure 7. Finite-element mesh with 3996 triangular elements.
Figure 8. Boundary conditions used in the FEMcalculations.
varied randomly for all grains within one calculation. Thus, although all grains had identical
properties with respect to their own local coordinate system, for each grain the orientation of
the local coordinate system was randomly chosen with respect to the reference frame e∼
. The
interfaces between adjoining grains (the grain boundaries) were not explicitly modelled, but
were merely assumed to be borders of grains over which the orientation of the anisotropic
elastic properties changes.
From the outcome of the finite-element calculations, we determined the effective elastic
constants that would yield the same average normal stresses along the edges as observed in the
finite-element calculation if the material were considered to be isotropic. In order to obtain ameasure for the average stresses and strains, the boundary conditions (which are represented
in figure 8) were chosen such that the edges of the domain remained straight. At the upper,
lower and left-hand sides the displacements in the normal direction were suppressed, while
at the right-hand side the displacements in the normal direction were set to a certain positive
value. The average stress component σ 11 was found by averaging the stress component σ 11
along the left-hand edge. The σ 22 stress component was obtained by averaging the σ 22 value
First we will discuss the results for the plain strain calculations. A total of 30 different sets of
56 random grain orientations were generated. For each previously described material, FEM
effective elastic constants were determined using equation (22) for each set of orientations andwere subsequently averaged with equation (19). In table 3 the resulting mean elastic constants
are summarized. The results are given in figures 9 and 10. In these figures each ‘◦’ marks
the FEM effective elastic constant resulting from one distinct set of orientations. The mean
and standard deviation of all 30 calculations are given for each material, as are the analytically
determined macroscopic effective elastic constants from section 2.
For the barium titanate material with a cubic structure and for zircon, the average FEM
effective constants virtually equal the Voigt–Reuss–Hill (VRH) value, which was assumed to
materials. For the other three materials, the average νσ FEM is not within the range between
the upper and lower bounds. The average effective Young’s modulus for tetragonal barium
titanate matches the VRH value, while for the remaining two materials the average E σ FEM is
considerably lower, but within the Voigt and Reuss bounds.
4.4. Discussion
For the different calculations with distinct sets of orientations, the effective elastic behaviour
of the simulated microstructures shows substantial scatter around a mean value. The effective
properties are determined by the exact orientations, and hence the specific properties of
the microstructures determine the mechanical behaviour on the scale considered. The most
important message from the computations, therefore, is that for relatively small polycrystalline
aggregates the microstructure must be explicitly taken into account in mechanical studies, andthe use of effective isotropic elastic properties is not correct. This confirms the conclusions
drawn by Mullen et al [3], who carried out similar computations.
It was expected that the average of the effective elastic behaviour of many small
microstructures would lead to the theoretical average behaviour explained in section 2, i.e.
that the mean value of the scattered point in the previous figures would correspond to the
Voigt–Reuss–Hill average. For some of our results, this is clearly not the case. Several
possible explanations for this unexpected behaviour can be given.
First, as we show in appendix A, the tetragonal barium titanate and the zircon material
properties may show nearly incompressible material behaviour for certain stress states. For
plane strain calculations, this may result in an overestimation of the elastic stiffness due to
mesh-locking (Hughes [15,p 208]). This may lead to the observed deviations.
Furthermore, a plane strain calculation forces the local strains in the e3 direction to be
zero everywhere, while for the calculation of the effective constants only global restrictions
are required, i.e. we have
ε33 = 0 instead of ε33 = 0
ε23 = 0 instead of ε23 = 0 (25)
ε31 = 0 instead of ε31 = 0.
However, this may lead to an overestimation of the effective constants for polycrystalline
aggregates for all anisotropic materials.
Another possible explanation for the average effective Young’s modulus being relativelyhigh for some plane strain calculations andrelatively low for most plane stresscalculations may
be the two-dimensional nature of the analysis. The analytically derived approximations for the
macroscopic effective elastic constants are based on a three-dimensional configuration of many
randomly oriented grains. However, the numerical approximations for the effective elastic
constants are based on the analysis of a two-dimensional configuration of (three-dimensionally
oriented) grains. Furthermore, boundary effects may be present and may influence the observed
In this paper we examined the relation between the (anisotropic) properties of the individual
grains and the effective macroscopic elastic behaviour of polycrystalline materials. For large
volumes of material, the conventional analytical theory of averaging the crystal properties canbe used to establish this relationship. We illustrated this with measurements of barium titanates
at various temperatures.
However, for relatively small volumes of materials with dimensions of the order of the
grain size, such as thin polycrystalline sheets, this analytical theory is no longer appropriate.
Therefore, we carried out micromechanical computations using a finite-element model to study
such structures. Our numerical model allows the simulation of the elastic properties of a real
microstructure, based on a microscopic image of the grain structure.
For various cubic and tetragonal materials the effective elastic properties were determined.
The results indeed turned out to be dependent on the precise orientations of the individual
grains. Hence, our main conclusion is that for relatively small polycrystalline aggregates, the
microstructure must be explicitly taken into account in mechanical studies, and the use of
average isotropic elastic properties is not correct.The computational results we presented here may be considered a first step towards a more
comprehensive modelling of the general relation between the microstructure of a material and
its overall effective mechanical properties. Ongoing work is directed towards simulation of
fracture, and the modelling of internal stresses between the grains that appear during sintering
of the microstructure.
Acknowledgments
The pulse excitation set-up was designed by M H M Rongen and J P van den Brink
(Philips Research Laboratories). S Oostra (Philips Central Development Passive Components)
prepared the barium titanate specimens. Figure 1 was provided by H Nabben (Philips Central
Development Passive Components), and figure 5 by R Apetz (Philips Research Laboratories).
Appendix A. Orthotropic properties of the materials studied
For certain stress states, the tetragonal barium titanate and the zircon material may show nearly
incompressible behaviour. This can be demonstrated by their orthotropic Poisson’s ratios.
We used the FEM package MARC to carry out the calculations. The following element
formulations were used. Details of the elements may be found in [14].
For the plane strain calculations, MARC element 125 was used. This is a second-orderisoparametric two-dimensional plane strain triangular element with nodes at the three corners
and the three midsides. A disadvantage of this element is that distortion during solution may
cause poor results. Therefore, as described in section 4.4, we also used element 11 for the plane
strain calculations. This is a linear four-node isoparametric plane strain quadrilateral element.
Although it is less sensitive to element distortions than element 125, shear behaviour may be
poorly represented. Theshear behaviour may be improved by using an alternative interpolation
function, such as the assumed strain procedure, which we also used. For nearly incompressible
behaviour, use of element 11 may lead to element locking, which can be eliminated with use
of the constant dilatation method, which we also applied to our problem (see section 4.4).
In the case of the plane stress calculations, we applied element 124, which is a quadratic
six-node plane stress triangle. Distortion of the element could lead to poor results, and a
quadrilateral element would be preferable in that case. Since we concentrate on the planestrain calculations in this paper, results with the quadrilateral plane stress element are not
reported here.
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